Consider a commutative noetherian local ring $R$ of dimension $d$ and define

$$c_R\colon=\min_{(x_1,\ldots,x_d)} \{\mathrm{length}\ R/(x_1,\ldots,x_d)R\mid (x_1,\ldots,x_d)\ \mathrm{is\ a\ system\ of parameters\ of\ }R\}.$$

1. Does $c_R$ always give information about the ring $R$? For instance, clearly $c_R=1$ if and only if $R$ is regular. When is $c_R=2$? Does $R$ have to be any particular ring for $c_R$ to be $2$?

2. In a given local ring $R$ can one characterize those system of parameters $(x_1,\ldots,x_d)$ for which $c_R=\mathrm{length}\ R/(x_1,\ldots,x_d)R$?